First slide

Methods of integration

Question

$Evaluate \int \frac{e^{x}}{e^{2 x} + 6 e^{x} + 5} d x$

Easy

A

$\frac{1}{4} \log |\frac{e^{x} + 1}{e^{x} + 5}| + C$
B

$\frac{1}{4} \log |\frac{e^{x} + 2}{e^{x} + 4}| + C$
C

$\frac{1}{4} \log |\frac{e^{x} + 5}{e^{x} + 1}| + C$
D

None of these

Solution

$I = \int \frac{e^{x}}{e^{2 t} + 6 e^{x} + 5} d x = \int \frac{e^{x}}{{(e^{x})}^{2} + 6 e^{x} + 5} d x$
$\begin{array}{l} Let e^{x} = t \Rightarrow e^{x} d x = d t \\ ∴ I = \int \frac{d t}{t^{2} + 6 t + 5} = \int \frac{1}{(t + 3)^{2} - 2^{2}} d t \\ = \frac{1}{2 \cdot 2} \log |\frac{t + 3 - 2}{t + 3 + 2}| + C = \frac{1}{4} \log |\frac{e^{x} + 1}{e^{x} + 5}| + C \end{array}$

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